Further Mathematics Questions and Answers
Test your knowledge of advanced level mathematics with this aptitude test. This test comprises Further Maths questions and answers from past JAMB and WAEC examinations.
Test your knowledge of advanced level mathematics with this aptitude test. This test comprises Further Maths questions and answers from past JAMB and WAEC examinations.
Given that \(f : x \to \frac{2x - 1}{x + 2}, x \neq -2\), find \(f^{-1}\), the inverse of f
\(f^{-1} : x \to \frac{1+2x}{2-x}, x \neq 2\)
\(f^{-1} : x \to \frac{1-2x}{x+2}, x \neq -2\)
\(f^{-1} : x \to \frac{1-2x}{x-2}, x \neq 2\)
\(f^{-1} : x \to \frac{1+2x}{x+2}, x \neq -2\)
Correct answer is A
\(f(x) = \frac{2x - 1}{x + 2}\)
\(y = \frac{2x - 1}{x + 2}\)
\(x = \frac{2y - 1}{y + 2} \implies x(y + 2) = 2y - 1\)
\(xy - 2y = -1 - 2x \implies y = \frac{-1 - 2x}{x - 2}\)
\(f^{-1} : x \to \frac{1 + 2x}{2 - x} ; x \neq 2\)
Which of the following is a factor of the polynomial \(6x^{4} + 2x^{3} + 15x + 5\)?
3x + 1
x + 1
2x + 1
x + 2
Correct answer is A
Using the remainder theorem, if (x - a) is a factor of f(x), then f(a) = 0.
Check the options and get the answer.
\(\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)
\(\begin{pmatrix} 27 & 12 \\ 16 & -15 \end{pmatrix}\)
\(\begin{pmatrix} -20 & -6 \\ 12 & -8 \end{pmatrix}\)
\(\begin{pmatrix} 11 & 12 \\ 30 & -11 \end{pmatrix}\)
Correct answer is D
\(P = \begin{pmatrix} -2 & 1 \\ 3 & 4 \end{pmatrix}; Q = \begin{pmatrix} 5 & -3 \\ 2 & -1 \end{pmatrix}\)
= \(PQ = \begin{pmatrix} -10+2 & 6-1 \\ 15+8 & -9-4 \end{pmatrix}\)
= \(\begin{pmatrix} -8 & 5 \\ 23 & -13 \end{pmatrix}\)
\(QP = \begin{pmatrix} -10-9 & 5-12 \\ -4-3 & 2-4 \end{pmatrix}\)
= \(\begin{pmatrix} -19 & -7 \\ -7 & -2 \end{pmatrix}\)
\(PQ - QP = \begin{pmatrix} -8 & 5 \\ 23 & -13 \end{pmatrix} - \begin{pmatrix} -19 & -7 \\ -7 & -2 \end{pmatrix}\)
= \(\begin{pmatrix} 11 & 12 \\ 30 & -11 \end{pmatrix}\)
Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\), find P and Q.
P = 4 and Q = 2
P = 2 and Q = 4
P = 4 and Q = -2
P = -2 and Q = 4
Correct answer is C
\(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\)
\(\frac{2x}{(x + 6)(x + 3)} = \frac{P(x + 3) + Q(x + 6)}{(x + 6)(x + 3)}\)
Comparing equations, we have
\(2x = Px + 3P + Qx + 6Q\)
\(\implies 3P + 6Q = 0 ... (1) ; P + Q = 2 .... (2)\)
From equation (1), \(3P = -6Q \implies P = -2Q\)
\(\therefore -2Q + Q = -Q = 2 \)
\(Q = -2\)
\(P = -2Q = -2(-2) = 4\)
\(P = 4, Q = -2\)
Given that \(f : x \to x^{2}\) and \(g : x \to x + 3\), where \(x \in R\), find \(f o g(2)\).
25
9
7
5
Correct answer is A
\(f : x \to x^{2} ; g : x \to x + 3\)
\(g(2) = 2 + 3 = 5\)
\(f o g(2) = f(5) = 5^{2} = 25\)